If X \sim P_o(\lambda) \quad then P(X=x) = \displaystyle \frac{e^{-\lambda} \lambda^x}{x!}
The classic Poisson example is the data set of von Bortkiewicz (1898), for the chance of a Prussian cavalryman being killed by the kick of a horse. Ten army corps were observed over 20 years, giving a total of 200 observations of one corps for a one year period. The period or module of observation is thus one year. The total deaths from horse kicks were 122, and the average number of deaths per year per corps was thus 122/200 = 0.61. This is a rate of less than 1. It is also obvious that it is meaningless to ask how many times per year a cavalryman was not killed by the kick of a horse. In any given year, we expect to observe, well, not exactly 0.61 deaths in one corps (that is not possible; deaths occur in modules of 1), but sometimes none, sometimes one, occasionally two, perhaps once in a while three, and (we might intuitively expect) very rarely any more. Here, then, is the classic Poisson situation: a rare event, whose average rate is small, with observations made over many small intervals of time.